Integers to Words (and Vice Versa)

“Two hundred” means 2 in the hundreds column, which gives us 200. “And six” means 6 in the ones column. The total is 200 + 6 = 206. The zero in the tens place is a placeholder — it shows there are no tens, but it’s essential for keeping each digit in the right column.

A student who writes 2006 is most likely using the “concatenation” strategy — writing “200” for “two hundred” and then sticking “6” on the end. But number names describe addition, not joining: 200 + 6 = 206, not “200” followed by “6.” Notice that 2006 would actually be read “two thousand and six” — a completely different number.

“Twelve hundred” means 12 × 100 = 1,200. “One thousand two hundred” means 1,000 + 200 = 1,200. Both give exactly 1,200.

Some students believe these are different numbers because “twelve hundred” sounds bigger — after all, twelve is bigger than one! But “twelve hundred” is an informal shortcut that groups the thousands and hundreds together, reading 1,200 as “12 hundreds” rather than “1 thousand and 2 hundreds.” Both partition the same value; the formal version simply uses the largest place-value word available.

In 4,008, the 8 sits in the ones column — the hundreds and tens columns both contain zero. “Four thousand and eight” = 4,000 + 8 = 4,008. If the answer were “four thousand and eighty,” the 8 would need to be in the tens column, giving 4,080.

The confusion arises from the way zero placeholders work. The two zeros in 4,008 hold the hundreds and tens places empty. Removing either zero would shift the 8 into a different column and change the number’s value entirely.

Break the name into additive parts: “twenty-three thousand” = 23 × 1,000 = 23,000. “Four hundred” = 400. “And five” = 5. Adding these gives 23,000 + 400 + 5 = 23,405.

A student who writes 234,005 is using the concatenation error — writing “23” then “400” then “5” side by side. But number names describe addition, not joining. Each word group fills specific place-value columns: 23 fills the ten-thousands and thousands, 4 fills the hundreds, 0 fills the tens, and 5 fills the ones.

Words in a number’s name are only used to describe columns with non-zero quantities. The word “zero” is entirely silent in standard place-value naming.

When there are consecutive zeros in the middle of a number (like the hundreds and tens in 5,004), we simply skip over them and join the non-zero parts together. We say “five thousand” and then jump straight to the “and four”, mentally relying on the gap in place-value labels to tell us where the zeros live.

Example: 3,005 — “three thousand and five”

Another: 2,060 — “two thousand and sixty”

Creative: 9,009 — “nine thousand and nine,” where the repeated 9 sits in completely different columns. Or 1,000 — “one thousand,” with no hundreds, tens, or ones at all.

Trap: 4,500 — “four thousand five hundred.” This DOES contain “hundred” because the hundreds digit is 5. To avoid “hundred,” the hundreds digit must be zero.

Example: 13 (thirteen) and 30 (thirty)

Another: 15 (fifteen) and 50 (fifty)

Creative: 18,000 (eighteen thousand) and 80,000 (eighty thousand) — the teen/ty confusion amplified by thousands, creating a difference of 62,000.

Trap: 21 and 12 — these use the same digits and are sometimes confused, but “twenty-one” and “twelve” don’t actually sound alike at all. The confusion is visual (same digits reversed), not auditory.

Example: 10,000 — “ten thousand” (5 digits, 2 words)

Another: 200 — “two hundred” (3 digits, 2 words)

Creative: 1,000,000 — “one million” (7 digits, 2 words). The gap between digits and words grows dramatically with round powers of ten.

Trap: 7 — “seven” (1 digit, 1 word). Equal, not more. The simplest numbers have the same number of digits and words, so they don’t satisfy the condition.

Example: 3,003 — “three thousand and three”

Another: 505 — “five hundred and five”

Creative: 111,111 — “one hundred and eleven thousand, one hundred and eleven.” The words “one,” “hundred,” and “eleven” each appear twice!

Trap: 1,210 — “one thousand, two hundred and ten.” Even though the digit 1 appears twice, each 1 translates to a different word (“one” in the thousands and “ten” at the end). No number-word is repeated.

Every three-digit number has a non-zero digit in the hundreds column, so its name always begins with “[digit] hundred.” This is true even for round hundreds: 100 = “one hundred,” 200 = “two hundred,” and so on through 900.

Students might try 99 or 1,000 as counterexamples, but 99 has only two digits and 1,000 has four — both fall outside the range. The three-digit range is defined by having a hundreds column, so the word “hundred” will always appear.

For 345, it does not work: the digit 3 is represented by two words (“three hundred”). Words like “hundred” or “thousand” represent column headers, not the digits themselves.

It also fails for numbers 11–19, which use special words. In 15, the digits are 1 and 5, but the word form is just “fifteen” — a single word for two digits. However, a student might reason it works for numbers like 40 (“forty” -> one word, one non-zero digit 4).

It’s true for 105 (“one hundred and five”) which has a zero in the tens column, and for 3,012 (“three thousand and twelve”) which has a zero in the hundreds column. In both cases, the “and” bridges a gap where one or more place-value columns are empty.

But it fails for 123 (“one hundred and twenty-three”) which has no zeros at all. In British English, “and” is routinely placed after the hundreds — whether or not there are empty columns. The word “and” is a linguistic convention, not a mathematical signal for zero.

Zero digits are always silent in standard number names. 10 is “ten,” not “one-zero.” 105 is “one hundred and five,” not “one hundred, zero, and five.” 4,008 is “four thousand and eight” — the two zeros in the hundreds and tens columns go completely unspoken.

This is why zero placeholders cause so many errors when converting between words and digits. Because zeros are never said aloud, students must infer their presence from gaps in the place-value structure. A student hearing “four thousand and eight” must recognise that the jump from thousands straight to ones means the hundreds and tens are both zero.

The student used a “concatenation” strategy — joining digit groups side by side instead of combining values by place value. They treated “three thousand” and “twelve” as separate chunks of digits to be stuck together, creating a six-digit number.

The correct approach: “three thousand” = 3,000 and “twelve” = 12. These are added, not joined: 3,000 + 12 = 3,012. Writing numbers isn’t like parking cars side-by-side in a lot (concatenation). It’s like stacking nested cups. The 12 sits inside the empty zeros of the 3,000.

The answer is correct — 450 is “four hundred and fifty” — but the “say each digit with its column name” method is unreliable and only works by luck here. It breaks down badly for numbers 11–19, which have irregular names.

For example, applying the method to 415 would give “four hundred, ten, five” or “four hundred, one-ty five” — neither of which is correct. The right answer is “four hundred and fifteen.” Similarly, 11 would become “ten-one” instead of “eleven.” The correct approach requires knowing the special names for numbers 11–19, not just mechanically reading digit by digit.

The student jumped over the zero placeholder and read the 6 and 5 as if they were next to each other. In 2,605, the 6 is in the hundreds column, the 0 is in the tens column, and the 5 is in the ones column. Skipping the zero changed which column each digit belongs in.

The correct answer is “two thousand, six hundred and five.” The student’s answer — “two thousand and sixty-five” — would be written as 2,065, a completely different number worth 540 less. Every zero must be accounted for, even if it isn’t spoken aloud.

The student confused “fourteen” with “forty.” 140 is “one hundred and forty,” not “one hundred and fourteen.” This is the classic teen/ty confusion: “fourteen” means 4 ones added to 10 (= 14), while “forty” means 4 tens (= 40).

The correct answer is 114 — “one hundred and fourteen.” The student’s approach of “taking 14 and adding a zero” actually multiplied 14 by 10, producing 140 (forty) instead of adding 14 to 100 to get 114 (fourteen).

The student has correctly identified the column, but they have incorrectly pluralized the place-value word. The correct answer is “four thousand”.

In English math conventions, place value labels like “hundred”, “thousand”, and “million” act as adjectives describing the digit before them. We never pluralize adjectives in English — just like we say “four red cars” instead of “four reds cars,” we say “four thousand.”