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Let us develop a program to print out the powers of 2 from 0 to 80.

For more information, see the following.

• Counting

Powers of two:

No carry:

Carry:

We can start with a simple C program.

Here is the C code.
#include <stdio.h> int main() { const int POWERMAX1 = 80; int a; a = 1; for (int powerPos1=0; powerPos1 <= POWERMAX1; powerPos1++) { printf("2<span class="c%">d is %d\n",powerPos1,a); a *= 2; } return 0; }
What might be wrong with this program? Here is the output of the C code.
2d is 0 2d is 1 2d is 2 2d is 3 2d is 4 2d is 5 2d is 6 2d is 7 2d is 8 2d is 9 2d is 10 2d is 11 2d is 12 2d is 13 2d is 14 2d is 15 2d is 16 2d is 17 2d is 18 2d is 19 2d is 20 2d is 21 2d is 22 2d is 23 2d is 24 2d is 25 2d is 26 2d is 27 2d is 28 2d is 29 2d is 30 2d is 31 2d is 32 2d is 33 2d is 34 2d is 35 2d is 36 2d is 37 2d is 38 2d is 39 2d is 40 2d is 41 2d is 42 2d is 43 2d is 44 2d is 45 2d is 46 2d is 47 2d is 48 2d is 49 2d is 50 2d is 51 2d is 52 2d is 53 2d is 54 2d is 55 2d is 56 2d is 57 2d is 58 2d is 59 2d is 60 2d is 61 2d is 62 2d is 63 2d is 64 2d is 65 2d is 66 2d is 67 2d is 68 2d is 69 2d is 70 2d is 71 2d is 72 2d is 73 2d is 74 2d is 75 2d is 76 2d is 77 2d is 78 2d is 79 2d is 80

Unfortunately, the programming language C overflows after 30 bits, goes to the maximum negative integer, then to zero and then remains at zero. Nothing happens. No run-time error. No compile-time error. Nothing. C just keeps on going.

How can we address this problem?

In such cases, we need a new data representation and an new solution (algorithm and pseudo-code).

We will represent each decimal digit, 0 to 9 with an array element as follows. Call the array a.

•  a[0] is the digit representing the count of 10⁰.
•  a[1] is the digit representing the count of 10¹.
•  a[2] is the digit representing the count of 10².
•  a[3] is the digit representing the count of 10³.
• ... and so on

To print the representation, print the digits in the array a from high to low.

To double, mulitple the digit at position 0 by 2, and then, if greater than 10, carry to the next digit/array position.

Here is the C code.
#include <stdio.h> int main() { const int POWERMAX1 = 80; int a[POWERMAX1+1]; int aLen1; int aPos1; int carry1; a[0] = 1; aLen1 = 1; for (int powerPos1=0; powerPos1 <= POWERMAX1; powerPos1++) { printf("2<span class="c%">d is ",powerPos1); for (int aPos1=aLen1-1; aPos1 >= 0; aPos1--) { printf("%d",a[aPos1]); if ((aPos1 % 3) == 0) { if (aPos1 != 0) { printf(","); } } } printf("\n"); carry1 = 0; aPos1 = 0; while (aPos1 < aLen1) { a[aPos1] = 2*a[aPos1]+carry1; carry1 = 0; if (a[aPos1] > 9) { a[aPos1] -= 10; carry1 = 1; } aPos1++; } if (carry1 == 1) { aLen1 += 1; a[aLen1-1] = 1; } } return 0; }
Here is the output of the C code.
2d is 1 2d is 2 2d is 4 2d is 8 2d is 16 2d is 32 2d is 64 2d is 128 2d is 256 2d is 512 2d is 1,024 2d is 2,048 2d is 4,096 2d is 8,192 2d is 16,384 2d is 32,768 2d is 65,536 2d is 131,072 2d is 262,144 2d is 524,288 2d is 1,048,576 2d is 2,097,152 2d is 4,194,304 2d is 8,388,608 2d is 16,777,216 2d is 33,554,432 2d is 67,108,864 2d is 134,217,728 2d is 268,435,456 2d is 536,870,912 2d is 1,073,741,824 2d is 2,147,483,648 2d is 4,294,967,296 2d is 8,589,934,592 2d is 17,179,869,184 2d is 34,359,738,368 2d is 68,719,476,736 2d is 137,438,953,472 2d is 274,877,906,944 2d is 549,755,813,888 2d is 1,099,511,627,776 2d is 2,199,023,255,552 2d is 4,398,046,511,104 2d is 8,796,093,022,208 2d is 17,592,186,044,416 2d is 35,184,372,088,832 2d is 70,368,744,177,664 2d is 140,737,488,355,328 2d is 281,474,976,710,656 2d is 562,949,953,421,312 2d is 1,125,899,906,842,624 2d is 2,251,799,813,685,248 2d is 4,503,599,627,370,496 2d is 9,007,199,254,740,992 2d is 18,014,398,509,481,984 2d is 36,028,797,018,963,968 2d is 72,057,594,037,927,936 2d is 144,115,188,075,855,872 2d is 288,230,376,151,711,744 2d is 576,460,752,303,423,488 2d is 1,152,921,504,606,846,976 2d is 2,305,843,009,213,693,952 2d is 4,611,686,018,427,387,904 2d is 9,223,372,036,854,775,808 2d is 18,446,744,073,709,551,616 2d is 36,893,488,147,419,103,232 2d is 73,786,976,294,838,206,464 2d is 147,573,952,589,676,412,928 2d is 295,147,905,179,352,825,856 2d is 590,295,810,358,705,651,712 2d is 1,180,591,620,717,411,303,424 2d is 2,361,183,241,434,822,606,848 2d is 4,722,366,482,869,645,213,696 2d is 9,444,732,965,739,290,427,392 2d is 18,889,465,931,478,580,854,784 2d is 37,778,931,862,957,161,709,568 2d is 75,557,863,725,914,323,419,136 2d is 151,115,727,451,828,646,838,272 2d is 302,231,454,903,657,293,676,544 2d is 604,462,909,807,314,587,353,088 2d is 1,208,925,819,614,629,174,706,176

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